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NIST certified data
Marcin Wojdyr edited this page Apr 27, 2011
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9 revisions
National Institute of Standards and Technology (U.S.) provides datasets for assessment of software accuracy for a variety of statistical methods. 27 datasets are provided for nonlinear regression testing: http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml. This page describes how the program was tested and presents the results.
Fityk gives more accurate results than two other programs that we also tested. This is a bit surprising, because, like most of the curve fitting programs, it uses (by default) Levenberg-Marquardt method and double precision for floating point numbers. In all the cases sum of squared residuals (Residual Sum of Squares in StRD) is exactly the same as certified values, so it is not listed below.
Fityk can read the data directly from Misra1a.dat, automatically skipping description, but since the file has y in the first column and x in the second, we need to swap x and y. Also, we ensure that the standard deviation of all y's is 1:
@0 < Misra1a.dat Y=x, X=y, S=1
After swapping the axes, you may click the Zoom All button in the GUI (or
type plot [] in the CLI). Since this dataset has only a few points, it is
a good idea to increase the size of the points (in GUI) to make them more
visible.
The certified values in the NIST files are presented in the scientific format with 11 decimal places. It is possible to set the same format in Fityk:
set numeric_format='%.10E'
Now let's prepare the model with the initial values from Start 1:
define Misra1a(b1, b2) = b1*(1-exp(-b2*x)) F = Misra1a(~500, ~0.0001)
To fit the data with a very high precision we change the stopping criteria:
=-> set lm_stop_rel_change=1e-20 =-> fit Fitting 2 (of 2) parameters to 14 points ... In L-M method: lambda=1e+15 > 1e+15, stopped. levenberg_marquardt: 100 iterations, 101 evaluations, 0.01 s. of CPU time. WSSR: 1.2455138894E-01 (-99.9988%) =-> i errors Standard errors: $_1 = 2.3894212918E+02 +- 2.7070075241E+00 $_2 = 5.5015643181E-04 +- 7.2668688436E-06
If Fityk was compiled with the option to use the long double type for
calculations (USE_LONG_DOUBLE set to 1 in
fityk.h)
we get exactly the same numbers as the certified values.
With the double precision used (which is the default) the parameters and
standard deviations differ at 10th or 11th decimal place.
All the results shown below are obtained with Fityk 1.0.0 compiled
with (only) double precision.
| Parameter | Standard Deviation | |
| b1 certified | 2.3894212918E+02 | 2.7070075241E+00 |
| b1 fityk/double | 2.3894212923E+02 | 2.7070075253E+00 |
| b2 certified | 5.5015643181E-04 | 7.2668688436E-06 |
| b2 fityk/double | 5.5015643168E-04 | 7.2668688435E-06 |
define Chwirut2(b1, b2, b3) = exp(-b1*x)/(b2+b3*x) F = Chwirut2(~0.1, ~0.01, ~0.02)
Again, we get exactly the certified results using the long double type:
=-> i errors Standard errors: $_1 = 1.6657666537E-01 +- 3.8303286810E-02 $_2 = 5.1653291286E-03 +- 6.6621605126E-04 $_3 = 1.2150007096E-02 +- 1.5304234767E-03
And the only difference when using double is in b2 on 11th place.
| Parameter | Standard Deviation | |
| b1 certified | 1.6657666537E-01 | 3.8303286810E-02 |
| b1 fityk/double | 1.6657666537E-01 | 3.8303286810E-02 |
| b2 certified | 5.1653291286E-03 | 6.6621605126E-04 |
| b2 fityk/double | 5.1653291285E-03 | 6.6621605127E-04 |
| b3 certified | 1.2150007096E-02 | 1.5304234767E-03 |
| b3 fityk/double | 1.2150007096E-02 | 1.5304234767E-03 |
define Chwirut1(b1, b2, b3) = exp(-b1*x)/(b2+b3*x) F = Chwirut1(~0.1, ~0.01, ~0.02)
| Parameter | Standard Deviation | |
| b1 certified | 1.9027818370E-01 | 2.1938557035E-02 |
| b1 fityk/double | 1.9027818358E-01 | 2.1938557029E-02 |
| b2 certified | 6.1314004477E-03 | 3.4500025051E-04 |
| b2 fityk/double | 6.1314004468E-03 | 3.4500025053E-04 |
| b3 certified | 1.0530908399E-02 | 7.9281847748E-04 |
| b3 fityk/double | 1.0530908402E-02 | 7.9281847747E-04 |
define Lanczos3(b1, b2, b3, b4, b5, b6) = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x) F = Lanczos3(~1.2, ~0.3, ~5.6, ~5.5, ~6.5, ~7.6)
| Parameter | Standard Deviation | |
| b1 certified | 8.6816414977E-02 | 1.7197908859E-02 |
| b1 fityk/double | 8.6816415134E-02 | 1.7197908832E-02 |
| b2 certified | 9.5498101505E-01 | 9.7041624475E-02 |
| b2 fityk/double | 9.5498101593E-01 | 9.7041624152E-02 |
| b3 certified | 8.4400777463E-01 | 4.1488663282E-02 |
| b3 fityk/double | 8.4400777500E-01 | 4.1488663264E-02 |
| b4 certified | 2.9515951832E+00 | 1.0766312506E-01 |
| b4 fityk/double | 2.9515951842E+00 | 1.0766312488E-01 |
| b5 certified | 1.5825685901E+00 | 5.8371576281E-02 |
| b5 fityk/double | 1.5825685896E+00 | 5.8371576235E-02 |
| b6 certified | 4.9863565084E+00 | 3.4436403035E-02 |
| b6 fityk/double | 4.9863565087E+00 | 3.4436403007E-02 |
define Gauss1(b1, b2, b3, b4, b5, b6, b7, b8) = b1*exp(-b2*x) + b3*exp(-(x-b4)^2 / b5^2) + b6*exp(-(x-b7)^2 / b8^2) F = Gauss1(~97.0, ~0.009, ~100.0, ~65.0, ~20.0, ~70.0, ~178.0, ~16.5)
This is the first dataset that gives all 11 digits in all results the same as the NIST certified values.
| Parameter | Standard Deviation | |
| b1 certified | 9.8778210871E+01 | 5.7527312730E-01 |
| b1 fityk/double | 9.8778210871E+01 | 5.7527312730E-01 |
| b2 certified | 1.0497276517E-02 | 1.1406289017E-04 |
| b2 fityk/double | 1.0497276517E-02 | 1.1406289017E-04 |
| b3 certified | 1.0048990633E+02 | 5.8831775752E-01 |
| b3 fityk/double | 1.0048990633E+02 | 5.8831775752E-01 |
| b4 certified | 6.7481111276E+01 | 1.0460593412E-01 |
| b4 fityk/double | 6.7481111276E+01 | 1.0460593412E-01 |
| b5 certified | 2.3129773360E+01 | 1.7439951146E-01 |
| b5 fityk/double | 2.3129773360E+01 | 1.7439951146E-01 |
| b6 certified | 7.1994503004E+01 | 6.2622793913E-01 |
| b6 fityk/double | 7.1994503004E+01 | 6.2622793913E-01 |
| b7 certified | 1.7899805021E+02 | 1.2436988217E-01 |
| b7 fityk/double | 1.7899805021E+02 | 1.2436988217E-01 |
| b8 certified | 1.8389389025E+01 | 2.0134312832E-01 |
| b8 fityk/double | 1.8389389025E+01 | 2.0134312832E-01 |
define Gauss2(b1, b2, b3, b4, b5, b6, b7, b8) = b1*exp(-b2*x) + b3*exp(-(x-b4)^2 / b5^2) + b6*exp(-(x-b7)^2 / b8^2) F = Gauss2(~97.0, ~0.009, ~100.0, ~65.0, ~20.0, ~70.0, ~178.0, ~16.5)
| Parameter | Standard Deviation | |
| b1 certified | 9.9018328406E+01 | 5.3748766879E-01 |
| b1 fityk/double | 9.9018328406E+01 | 5.3748766879E-01 |
| b2 certified | 1.0994945399E-02 | 1.3335306766E-04 |
| b2 fityk/double | 1.0994945399E-02 | 1.3335306766E-04 |
| b3 certified | 1.0188022528E+02 | 5.9217315772E-01 |
| b3 fityk/double | 1.0188022528E+02 | 5.9217315773E-01 |
| b4 certified | 1.0703095519E+02 | 1.5006798316E-01 |
| b4 fityk/double | 1.0703095519E+02 | 1.5006798316E-01 |
| b5 certified | 2.3578584029E+01 | 2.2695595067E-01 |
| b5 fityk/double | 2.3578584028E+01 | 2.2695595067E-01 |
| b6 certified | 7.2045589471E+01 | 6.1721965884E-01 |
| b6 fityk/double | 7.2045589471E+01 | 6.1721965883E-01 |
| b7 certified | 1.5327010194E+02 | 1.9466674341E-01 |
| b7 fityk/double | 1.5327010194E+02 | 1.9466674342E-01 |
| b8 certified | 1.9525972636E+01 | 2.6416549393E-01 |
| b8 fityk/double | 1.9525972637E+01 | 2.6416549394E-01 |
define DanWood(b1, b2) = b1*x^b2 F = DanWood(~1, ~5)
All digits the same.
| Parameter | Standard Deviation | |
| b1 certified | 7.6886226176E-01 | 1.8281973860E-02 |
| b1 fityk/double | 7.6886226176E-01 | 1.8281973860E-02 |
| b2 certified | 3.8604055871E+00 | 5.1726610913E-02 |
| b2 fityk/double | 3.8604055871E+00 | 5.1726610913E-02 |
define Misra1b(b1, b2) = b1 * (1-(1+b2*x/2)^(-2)) F = Misra1b(~500, ~0.0001)
Again, we get exactly the certified results.
| Parameter | Standard Deviation | |
| b1 certified | 3.3799746163E+02 | 3.1643950207E+00 |
| b1 fityk/double | 3.3799746163E+02 | 3.1643950207E+00 |
| b2 certified | 3.9039091287E-04 | 4.2547321834E-06 |
| b2 fityk/double | 3.9039091287E-04 | 4.2547321834E-06 |
define Kirby2(b1, b2, b3, b4, b5) = (b1 + b2*x + b3*x^2) / (1 + b4*x + b5*x^2) F = Kirby2(~2, ~-0.1, ~0.003, ~-0.001, ~0.00001)
| Parameter | Standard Deviation | |
| b1 certified | 1.6745063063E+00 | 8.7989634338E-02 |
| b1 fityk/double | 1.6745063047E+00 | 8.7989634326E-02 |
| b2 certified | -1.3927397867E-01 | 4.1182041386E-03 |
| b2 fityk/double | -1.3927397859E-01 | 4.1182041376E-03 |
| b3 certified | 2.5961181191E-03 | 4.1856520458E-05 |
| b3 fityk/double | 2.5961181182E-03 | 4.1856520445E-05 |
| b4 certified | -1.7241811870E-03 | 5.8931897355E-05 |
| b4 fityk/double | -1.7241811882E-03 | 5.8931897332E-05 |
| b5 certified | 2.1664802578E-05 | 2.0129761919E-07 |
| b5 fityk/double | 2.1664802573E-05 | 2.0129761914E-07 |
define Hahn1(b1, b2, b3, b4, b5, b6, b7) = (b1+b2*x+b3*x^2+b4*x^3) / (1+b5*x+b6*x^2+b7*x^3) F = Hahn1(~10, ~-1, ~0.05, ~-0.00001, ~-0.05, ~0.001, ~-0.000001)
| Parameter | Standard Deviation | |
| b1 certified | 1.0776351733E+00 | 1.7070154742E-01 |
| b1 fityk/double | 1.0776351730E+00 | 1.7070154741E-01 |
| b2 certified | -1.2269296921E-01 | 1.2000289189E-02 |
| b2 fityk/double | -1.2269296919E-01 | 1.2000289189E-02 |
| b3 certified | 4.0863750610E-03 | 2.2508314937E-04 |
| b3 fityk/double | 4.0863750606E-03 | 2.2508314935E-04 |
| b4 certified | -1.4262662514E-06 | 2.7578037666E-07 |
| b4 fityk/double | -1.4262662508E-06 | 2.7578037665E-07 |
| b5 certified | -5.7609940901E-03 | 2.4712888219E-04 |
| b5 fityk/double | -5.7609940903E-03 | 2.4712888218E-04 |
| b6 certified | 2.4053735503E-04 | 1.0449373768E-05 |
| b6 fityk/double | 2.4053735501E-04 | 1.0449373767E-05 |
| b7 certified | -1.2314450199E-07 | 1.3027335327E-08 |
| b7 fityk/double | -1.2314450196E-07 | 1.3027335327E-08 |
Dataset Nelson has 2 predictors (x1 and x2), Fityk does not handle 3D data, we skip this test.
define MGH17(b1, b2, b3, b4, b5) = b1 + b2*exp(-x*b4) + b3*exp(-x*b5) F = MGH17(~0.5, ~1.5, ~-1, ~0.01, ~0.02)
| Parameter | Standard Deviation | |
| b1 certified | 3.7541005211E-01 | 2.0723153551E-03 |
| b1 fityk/double | 3.7541005211E-01 | 2.0723153551E-03 |
| b2 certified | 1.9358469127E+00 | 2.2031669222E-01 |
| b2 fityk/double | 1.9358469126E+00 | 2.2031669217E-01 |
| b3 certified | -1.4646871366E+00 | 2.2175707739E-01 |
| b3 fityk/double | -1.4646871365E+00 | 2.2175707733E-01 |
| b4 certified | 1.2867534640E-02 | 4.4861358114E-04 |
| b4 fityk/double | 1.2867534640E-02 | 4.4861358109E-04 |
| b5 certified | 2.2122699662E-02 | 8.9471996575E-04 |
| b5 fityk/double | 2.2122699662E-02 | 8.9471996569E-04 |
define Lanczos1(b1, b2, b3, b4, b5, b6) = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x) F = Lanczos1(~1.2, ~0.3, ~5.6, ~5.5, ~6.5, ~7.6)
| Parameter | Standard Deviation | |
| b1 certified | 9.5100000027E-02 | 5.3347304234E-11 |
| b1 fityk/double | 9.5100000027E-02 | 5.3291941815E-11 |
| b2 certified | 1.0000000001E+00 | 2.7473038179E-10 |
| b2 fityk/double | 1.0000000001E+00 | 2.7444527387E-10 |
| b3 certified | 8.6070000013E-01 | 1.3576062225E-10 |
| b3 fityk/double | 8.6070000013E-01 | 1.3561973349E-10 |
| b4 certified | 3.0000000002E+00 | 3.3308253069E-10 |
| b4 fityk/double | 3.0000000002E+00 | 3.3273686649E-10 |
| b5 certified | 1.5575999998E+00 | 1.8815731448E-10 |
| b5 fityk/double | 1.5575999998E+00 | 1.8796204982E-10 |
| b6 certified | 5.0000000001E+00 | 1.1057500538E-10 |
| b6 fityk/double | 5.0000000001E+00 | 1.1046025358E-10 |
define Lanczos2(b1, b2, b3, b4, b5, b6) = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x) F = Lanczos2(~1.2, ~0.3, ~5.6, ~5.5, ~6.5, ~7.6)
| Parameter | Standard Deviation | |
| b1 certified | 9.6251029939E-02 | 6.6770575477E-04 |
| b1 fityk/double | 9.6251029939E-02 | 6.6770575433E-04 |
| b2 certified | 1.0057332849E+00 | 3.3989646176E-03 |
| b2 fityk/double | 1.0057332849E+00 | 3.3989646154E-03 |
| b3 certified | 8.6424689056E-01 | 1.7185846685E-03 |
| b3 fityk/double | 8.6424689056E-01 | 1.7185846674E-03 |
| b4 certified | 3.0078283915E+00 | 4.1707005856E-03 |
| b4 fityk/double | 3.0078283915E+00 | 4.1707005829E-03 |
| b5 certified | 1.5529016879E+00 | 2.3744381417E-03 |
| b5 fityk/double | 1.5529016879E+00 | 2.3744381401E-03 |
| b6 certified | 5.0028798100E+00 | 1.3958787284E-03 |
| b6 fityk/double | 5.0028798100E+00 | 1.3958787275E-03 |
define Gauss3(b1, b2, b3, b4, b5, b6, b7, b8) = b1*exp(-b2*x) + b3*exp(-(x-b4)^2 / b5^2) + b6*exp(-(x-b7)^2 / b8^2) F = Gauss3(~97.0, ~0.009, ~100.0, ~65.0, ~20.0, ~70.0, ~178.0, ~16.5)
| Parameter | Standard Deviation | |
| b1 certified | 9.8940368970E+01 | 5.3005192833E-01 |
| b1 fityk/double | 9.8940368970E+01 | 5.3005192832E-01 |
| b2 certified | 1.0945879335E-02 | 1.2554058911E-04 |
| b2 fityk/double | 1.0945879335E-02 | 1.2554058910E-04 |
| b3 certified | 1.0069553078E+02 | 8.1256587317E-01 |
| b3 fityk/double | 1.0069553078E+02 | 8.1256587330E-01 |
| b4 certified | 1.1163619459E+02 | 3.5317859757E-01 |
| b4 fityk/double | 1.1163619459E+02 | 3.5317859756E-01 |
| b5 certified | 2.3300500029E+01 | 3.6584783023E-01 |
| b5 fityk/double | 2.3300500027E+01 | 3.6584783020E-01 |
| b6 certified | 7.3705031418E+01 | 1.2091239082E+00 |
| b6 fityk/double | 7.3705031424E+01 | 1.2091239080E+00 |
| b7 certified | 1.4776164251E+02 | 4.0488183351E-01 |
| b7 fityk/double | 1.4776164250E+02 | 4.0488183353E-01 |
| b8 certified | 1.9668221230E+01 | 3.7806634336E-01 |
| b8 fityk/double | 1.9668221231E+01 | 3.7806634338E-01 |
define Misra1c(b1, b2) = b1 * (1-(1+2*b2*x)^(-.5)) F = Misra1c(~500, ~0.0001)
| Parameter | Standard Deviation | |
| b1 certified | 6.3642725809E+02 | 4.6638326572E+00 |
| b1 fityk/double | 6.3642725809E+02 | 4.6638326572E+00 |
| b2 certified | 2.0813627256E-04 | 1.7728423155E-06 |
| b2 fityk/double | 2.0813627256E-04 | 1.7728423155E-06 |
define Misra1d(b1, b2) = b1*b2*x*((1+b2*x)^(-1)) F = Misra1d(~500, ~0.0001)
| Parameter | Standard Deviation | |
| b1 certified | 4.3736970754E+02 | 3.6489174345E+00 |
| b1 fityk/double | 4.3736970754E+02 | 3.6489174345E+00 |
| b2 certified | 3.0227324449E-04 | 2.9334354479E-06 |
| b2 fityk/double | 3.0227324449E-04 | 2.9334354479E-06 |
define Roszman1(b1, b2, b3, b4) = b1 - b2*x - atan(b3/(x-b4))/pi F = Roszman1(~0.1, ~-0.00001, ~1000, ~-100)
| Parameter | Standard Deviation | |
| b1 certified | 2.0196866396E-01 | 1.9172666023E-02 |
| b1 fityk/double | 2.0196866395E-01 | 1.9172666023E-02 |
| b2 certified | -6.1953516256E-06 | 3.2058931691E-06 |
| b2 fityk/double | -6.1953516245E-06 | 3.2058931692E-06 |
| b3 certified | 1.2044556708E+03 | 7.4050983057E+01 |
| b3 fityk/double | 1.2044556708E+03 | 7.4050983056E+01 |
| b4 certified | -1.8134269537E+02 | 4.9573513849E+01 |
| b4 fityk/double | -1.8134269538E+02 | 4.9573513852E+01 |
define Enso(b1, b2, b3, b4, b5, b6, b7, b8, b9) = b1 + b2*cos(t/12) + b3*sin(t/12) + b5*cos(t/b4) + b6*sin(t/b4) + b8*cos(t/b7) + b9*sin(t/b7) where t=2*pi*x F = Enso(~11.0, ~3.0, ~0.5, ~40.0, ~-0.7, ~-1.3, ~25.0, ~-0.3, ~1.4)
| Parameter | Standard Deviation | |
| b1 certified | 1.0510749193E+01 | 1.7488832467E-01 |
| b1 fityk/double | 1.0510749192E+01 | 1.7488832468E-01 |
| b2 certified | 3.0762128085E+00 | 2.4310052139E-01 |
| b2 fityk/double | 3.0762128089E+00 | 2.4310052139E-01 |
| b3 certified | 5.3280138227E-01 | 2.4354686618E-01 |
| b3 fityk/double | 5.3280138256E-01 | 2.4354686618E-01 |
| b4 certified | 4.4311088700E+01 | 9.4408025976E-01 |
| b4 fityk/double | 4.4311088678E+01 | 9.4408025781E-01 |
| b5 certified | -1.6231428586E+00 | 2.8078369611E-01 |
| b5 fityk/double | -1.6231428614E+00 | 2.8078369522E-01 |
| b6 certified | 5.2554493756E-01 | 4.8073701119E-01 |
| b6 fityk/double | 5.2554493176E-01 | 4.8073701153E-01 |
| b7 certified | 2.6887614440E+01 | 4.1612939130E-01 |
| b7 fityk/double | 2.6887614416E+01 | 4.1612939012E-01 |
| b8 certified | 2.1232288488E-01 | 5.1460022911E-01 |
| b8 fityk/double | 2.1232286294E-01 | 5.1460022934E-01 |
| b9 certified | 1.4966870418E+00 | 2.5434468893E-01 |
| b9 fityk/double | 1.4966870451E+00 | 2.5434468712E-01 |
define MGH09(b1, b2, b3, b4) = b1*(x^2+x*b2) / (x^2+x*b3+b4) F = MGH09(~25, ~39, ~41.5, ~39)
| Parameter | Standard Deviation | |
| b1 certified | 1.9280693458E-01 | 1.1435312227E-02 |
| b1 fityk/double | 1.9280693477E-01 | 1.1435312166E-02 |
| b2 certified | 1.9128232873E-01 | 1.9633220911E-01 |
| b2 fityk/double | 1.9128232455E-01 | 1.9633220578E-01 |
| b3 certified | 1.2305650693E-01 | 8.0842031232E-02 |
| b3 fityk/double | 1.2305650614E-01 | 8.0842030241E-02 |
| b4 certified | 1.3606233068E-01 | 9.0025542308E-02 |
| b4 fityk/double | 1.3606232875E-01 | 9.0025541048E-02 |
define Thurber(b1, b2, b3, b4, b5, b6, b7) = (b1 + b2*x + b3*x^2 + b4*x^3) / (1 + b5*x + b6*x^2 + b7*x^3) F = Thurber(~1000, ~1000, ~400, ~40, ~0.7, ~0.3, ~0.03)
| Parameter | Standard Deviation | |
| b1 certified | 1.2881396800E+03 | 4.6647963344E+00 |
| b1 fityk/double | 1.2881396799E+03 | 4.6647963334E+00 |
| b2 certified | 1.4910792535E+03 | 3.9571156086E+01 |
| b2 fityk/double | 1.4910792548E+03 | 3.9571155198E+01 |
| b3 certified | 5.8323836877E+02 | 2.8698696102E+01 |
| b3 fityk/double | 5.8323836966E+02 | 2.8698695459E+01 |
| b4 certified | 7.5416644291E+01 | 5.5675370270E+00 |
| b4 fityk/double | 7.5416644466E+01 | 5.5675368991E+00 |
| b5 certified | 9.6629502864E-01 | 3.1333340687E-02 |
| b5 fityk/double | 9.6629502951E-01 | 3.1333340084E-02 |
| b6 certified | 3.9797285797E-01 | 1.4984928198E-02 |
| b6 fityk/double | 3.9797285837E-01 | 1.4984927908E-02 |
| b7 certified | 4.9727297349E-02 | 6.5842344623E-03 |
| b7 fityk/double | 4.9727297607E-02 | 6.5842343110E-03 |
define BoxBOD(b1, b2) = b1*(1-exp(-b2*x)) F = BoxBOD(~100, ~0.75)
| Parameter | Standard Deviation | |
| b1 certified | 2.1380940889E+02 | 1.2354515176E+01 |
| b1 fityk/double | 2.1380940890E+02 | 1.2354515178E+01 |
| b2 certified | 5.4723748542E-01 | 1.0455993237E-01 |
| b2 fityk/double | 5.4723748534E-01 | 1.0455993235E-01 |
define Rat42(b1, b2, b3) = b1 / (1+exp(b2-b3*x)) F = Rat42(~100, ~1, ~0.1)
| Parameter | Standard Deviation | |
| b1 certified | 7.2462237576E+01 | 1.7340283401E+00 |
| b1 fityk/double | 7.2462237576E+01 | 1.7340283401E+00 |
| b2 certified | 2.6180768402E+00 | 8.8295217536E-02 |
| b2 fityk/double | 2.6180768402E+00 | 8.8295217536E-02 |
| b3 certified | 6.7359200066E-02 | 3.4465663377E-03 |
| b3 fityk/double | 6.7359200066E-02 | 3.4465663377E-03 |
define MGH10(b1, b2, b3) = b1 * exp(b2/(x+b3)) F = MGH10(~0.02, ~4000, ~250)
| Parameter | Standard Deviation | |
| b1 certified | 5.6096364710E-03 | 1.5687892471E-04 |
| b1 fityk/double | 5.6096364710E-03 | 1.5687892478E-04 |
| b2 certified | 6.1813463463E+03 | 2.3309021107E+01 |
| b2 fityk/double | 6.1813463463E+03 | 2.3309021118E+01 |
| b3 certified | 3.4522363462E+02 | 7.8486103508E-01 |
| b3 fityk/double | 3.4522363462E+02 | 7.8486103544E-01 |
define Eckerle4(b1, b2, b3) = b1 * exp(b2/(x+b3)) F = Eckerle4(~1.5, ~5, ~450)
| Parameter | Standard Deviation | |
| b1 certified | 1.5543827178E+00 | 1.5408051163E-02 |
| b1 fityk/double | 1.5543827179E+00 | 1.5408051165E-02 |
| b2 certified | 4.0888321754E+00 | 4.6803020753E-02 |
| b2 fityk/double | 4.0888321764E+00 | 4.6803020765E-02 |
| b3 certified | 4.5154121844E+02 | 4.6800518816E-02 |
| b3 fityk/double | 4.5154121844E+02 | 4.6800518828E-02 |
define Rat43(b1, b2, b3, b4) = b1 / ((1+exp(b2-b3*x))^(1/b4)) F = Rat43(~100, ~10, ~1, ~1)
| Parameter | Standard Deviation | |
| b1 certified | 6.9964151270E+02 | 1.6302297817E+01 |
| b1 fityk/double | 6.9964151266E+02 | 1.6302297809E+01 |
| b2 certified | 5.2771253025E+00 | 2.0828735829E+00 |
| b2 fityk/double | 5.2771253072E+00 | 2.0828735841E+00 |
| b3 certified | 7.5962938329E-01 | 1.9566123451E-01 |
| b3 fityk/double | 7.5962938380E-01 | 1.9566123468E-01 |
| b4 certified | 1.2792483859E+00 | 6.8761936385E-01 |
| b4 fityk/double | 1.2792483872E+00 | 6.8761936428E-01 |
define Bennett5(b1, b2, b3) = b1 * (b2+x)^(-1/b3) F = Bennett5(~-2000, ~50, ~0.8)
| Parameter | Standard Deviation | |
| b1 certified | -2.5235058043E+03 | 2.9715175411E+02 |
| b1 fityk/double | -2.5235058043E+03 | 2.9715175183E+02 |
| b2 certified | 4.6736564644E+01 | 1.2448871856E+00 |
| b2 fityk/double | 4.6736564644E+01 | 1.2448871760E+00 |
| b3 certified | 9.3218483193E-01 | 2.0272299378E-02 |
| b3 fityk/double | 9.3218483193E-01 | 2.0272299222E-02 |